Q. 94.3( 362 Votes )

# In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 8.20). Show that:

(i) Δ APD ≅Δ CQB

(ii) AP = CQ

(iii) Δ AQB ≅Δ CPD

(iv) AQ = CP

(v) APCQ is a parallelogram

Answer :

**(i)** In ΔAPD and ΔCQB,

∠ADP = ∠CBQ (Alternate interior angles for BC || AD)

AD = CB (Opposite sides of parallelogram ABCD)

DP = BQ (Given)

Two sides and included angle (**SAS**) Definition: Triangles are

**congruent**if any pair of corresponding sides and their included angles are equal in both triangles.

ΔAPD ΔCQB (Using SAS congruence rule)

**(ii)** As we had observed that,

ΔAPD ΔCQB

AP = CQ (CPCT)

**(iii)** In ΔAQB and ΔCPD,

∠ABQ = ∠CDP (Alternate interior angles for AB || CD)

AB = CD (Opposite sides of parallelogram ABCD)

BQ = DP (Given)

Two sides and included angle (**SAS**) Definition: Triangles are

**congruent**if any pair of corresponding sides and their included angles are equal in both triangles.

ΔAQB ΔCPD (Using SAS congruence rule)

**(iv)** As we had observed that,

ΔAQB ΔCPD,

AQ = CP (CPCT)

**(v)** From the result obtained in (ii) and (iv),

AQ = CP and AP = CQ

Since,

Opposite sides in quadrilateral APCQ are equal to each other,

APCQ is a parallelogram

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